Let $A$ and $B$ be two $p$-torsion free $\overline{\mathbb{Z}}_p$-algebras of finite type and suppose that $A\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{Q}}_{p}$ and $B\otimes_{\overline{\mathbb{Z}}_{p}}\overline{\mathbb{Q}}_{p}$ are isomorphic as $\overline{\mathbb{Q}}_{p}$-algebras.
In general we cannot conclude that $A$ and $B$ are isomorphic as $\overline{\mathbb{Z}}_p$-algebra (see example below). Under what conditions we can conclude that $A$ and $B$ have equivalent module categories?
If we take $A=\overline{\mathbb{Z}}_2[x]/(x^4-1)$ and $B=\overline{\mathbb{Z}}_2[y,z]/(y^2-1,z^2-1)$ then $A\otimes_{\overline{\mathbb{Z}}_{2}}\overline{\mathbb{Q}}_{2}$ and $B\otimes_{\overline{\mathbb{Z}}_{2}}\overline{\mathbb{Q}}_{2}$ are both isomorphic to $(\overline{\mathbb{Q}}_{2})^{64}$ but they are not isomorphic and they don't seem to have equivalent module categories.